A finite point method for adaptive three‐dimensional compressible flow calculations

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of a meshless solver for high Reynolds number flow

In this paper, the extension of an upwind least‐square based meshless solver to high Reynolds number flow is explored, and the properties of the meshless solver are analyzed both theoretically and numerically. Existing works have verified the meshless solver mostly with inviscid flows and low Reynolds number flows, and in this work, we are interested in the behavior of the meshless solver for high Reynolds number flow, especially in the near‐wall region. With both theoretical and numerical analysis, the effects of two parameters on the meshless solver are identified. The first one is the misalignment effect caused by the significantly skewed supporting points, and it is found that the meshless solver still yields accurate prediction. It is a very interesting property and is opposite to the median‐dual control volume based vertex‐centered finite volume method, which is known to give degraded result with stretched triangular/tetrahedral cells in the near‐wall region. The second parameter is the curvature, and according to theoretical analysis, it is found in the region with both large aspect ratio and curvature, and the streamwise residual is less affected; however, the wall‐normal counterpart suffers from accuracy degradation. In this paper, an improved method that uses a meshless solver for the streamwise residual and finite difference for wall‐normal residual is developed. This method is proved to be less sensitive to the curvature and provides improved accuracy. This work presents an understanding of the meshless solver for high Reynolds number flow computation, and the analysis in this paper is verified with a series of numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

An immersed boundary vector potential-vorticity meshless solver of the incompressible Navier–Stokes equation

We present a strong form meshless solver for numerical solution of the nonstationary, incompressible, viscous Navier–Stokes equations in two (2D) and three dimensions (3D). We solve the flow equations in their stream function-vorticity (in 2D) and vector potential-vorticity (in 3D) formulation, by extending to 3D flows the boundary condition-enforced immersed boundary method, originally introduced in the literature for 2D problems. We use a Cartesian grid, uniform or locally refined, to discretize the spatial domain. We apply an explicit time integration scheme to update the transient vorticity equations, and we solve the Poisson type equation for the stream function or vector potential field using the meshless point collocation method. Spatial derivatives of the unknown field functions are computed using the discretization-corrected particle strength exchange method. We verify the accuracy of the proposed numerical scheme through commonly used benchmark and example problems. Excellent agreement with the data from the literature was achieved. The proposed method was shown to be very efficient, having relatively large critical time steps.     

Kinetic meshless method for compressible flows

We present a grid‐free or meshless approximation called the kinetic meshless method (KMM), for the numerical solution of hyperbolic conservation laws that can be obtained by taking moments of a Boltzmann‐type transport equation. The meshless formulation requires the domain discretization to have very little topological information; a distribution of points in the domain together with local connectivity information is sufficient. For each node, the connectivity consists of a set of nearby nodes which are used to evaluate the spatial derivatives appearing in the conservation law. The derivatives are obtained using a modified form of the least‐squares approximation. The method is applied to the Euler equations for inviscid flow and results are presented for some 2‐D problems. The ability of the new scheme to accurately compute inviscid flows is clearly demonstrated, including good shock capturing ability. Comparisons with other grid‐free methods are made showing some advantages of the current approach. Copyright © 2007 John Wiley & Sons, Ltd.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

A fully coupled numerical technique for 2D bed morphology calculations

A fully coupled two‐dimensional subcritical and/or supercritical, viscous, free‐surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free‐surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite‐volume scheme using transformed grid in order to handle complex geometry fluvial problems. Convergence is accelerated with use of a multi‐grid technique. Firstly the capabilities of the proposed method are demonstrated by analyzing subcritical and supercritical hydrodynamic flows. Thereafter, an analysis of one‐ and two‐dimensional flows is performed referring to aggradation and scouring. For all reported test cases the computed results compare reasonably well with measurements as well as with other numerical solutions. The method is stable, reliable and accurate handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries. Copyright © 2002 John Wiley & Sons, Ltd.     

On the accuracy and efficiency of CFD methods in real gas hypersonics

A study of viscous and inviscid hypersonic flows using generalized upwind methods is presented. A new family of hybrid flux-splitting methods is examined for hypersonic flows. The hybrid method is constructed by the superposition of the flux-vector-splitting (FVS) method and second-order artificial dissipation in the regions of strong shock waves. The conservative variables on the cell faces are calculated by an upwind extrapolation scheme to third-order accuracy. A second-order-accurate scheme is used for the discretization of the viscous terms. The solution of the system of equations is achieved by an implicit unfactored method. In order to reduce the computational time, a local adaptive mesh solution (LAMS) method is proposed. The LAMS method combines the mesh-sequencing technique and local solution of the equations. The local solution of either the Euler or the NAVIER-STOKES equations is applied for the region of the flow field where numerical disturbances die out slowly. Validation of the Euler and NAVIER-STOKES codes is obtained for hypersonic flows around blunt bodies. Real gas effects are introduced via a generalized equation of state.     

A finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

A fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.     

Unified one‐fluid formulation for incompressible flexible solids and multiphase flows: Application to hydrodynamics using the immersed structural potential method (ISPM)

In this paper, we present a two‐dimensional computational framework for the simulation of fluid‐structure interaction problems involving incompressible flexible solids and multiphase flows, further extending the application range of classical immersed computational approaches to the context of hydrodynamics. The proposed method aims to overcome shortcomings such as the restriction of having to deal with similar density ratios among different phases or the restriction to solve single‐phase flows. First, a variation of classical immersed techniques, pioneered with the immersed boundary method (IBM), is presented by rearranging the governing equations, which define the behaviour of the multiple physics involved. The formulation is compatible with the “one‐fluid” formulation for two‐phase flows and can deal with large density ratios with the help of an anisotropic Poisson solver. Second, immersed deformable structures and fluid phases are modelled in an identical manner except for the computation of the deviatoric stresses. The numerical technique followed in this paper builds upon the immersed structural potential method developed by the authors, by adding a level set–based method for the capturing of the fluid‐fluid interfaces and an interface Lagrangian‐based meshless technique for the tracking of the fluid‐structure interface. The spatial discretisation is based on the standard marker‐and‐cell method used in conjunction with a fractional step approach for the pressure/velocity decoupling, a second‐order time integrator, and a fixed‐point iterative scheme. The paper presents a wide d range of two‐dimensional applications involving multiphase flows interacting with immersed deformable solids, including benchmarking against both experimental and alternative numerical schemes.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.     

Modified Lagrangian vortex method with improved boundary conditions for water waves past a thin bottom‐standing barrier

Herein, the modified Lagrangian vortex method (LVM), a hybrid analytical‐numerical algorithm per se, is devised to simulate the process of vortex formation and shedding from the sharp edge of a zero‐thickness vertical plate under linear water‐wave attack. Application of the Helmholtz decomposition facilitates a convenient switch between the inviscid‐ and viscous‐flow models, thereby enabling easy incorporation of vorticity effects into the potential‐flow calculations for the viscous‐dominated region. In evaluating the potential‐flow component, making good use of the quickly convergent technique with singular basis functions, correctly capturing the singular behavior in velocity fields near the tip of the plate, leads to a considerable reduction of computational burdens and to 12‐decimal‐place accuracy. The viscous correction is carried out via the meshless LVM with improved boundary conditions. Comparisons with previously published results show good agreement. Simulations of vortex generation and evolution illuminate the ability of the present method, and provide a supplement to pertinent experimental works. The hybrid scheme proposed herein allows flexibility for the former LVM and convenience in the code development. Such a compromise fits particularly well for the high‐resolution modeling of sharp‐edged vortex shedding without heavy numerical developments. Copyright © 2014 John Wiley & Sons, Ltd.     

Fictitious domain approach for numerical modelling of Navier–Stokes equations

This study investigates a fictitious domain model for the numerical solution of various incompressible viscous flows. It is based on the so‐called Navier–Stokes/Brinkman and energy equations with discontinuous coefficients all over an auxiliary embedding domain. The solid obstacles or walls are taken into account by a penalty technique. Some volumic control terms are directly introduced in the governing equations in order to prescribe immersed boundary conditions. The implicit numerical scheme, which uses an upwind finite volume method on staggered Cartesian grids, is of second‐order accuracy in time and space. A multigrid local mesh refinement is also implemented, using the multi‐level Zoom Flux Interface Correction (FIC) method, in order to increase the precision where it is needed in the domain. At each time step, some iterations of the augmented Lagrangian method combined with a preconditioned Krylov algorithm allow the divergence‐free velocity and pressure fields be solved for. The tested cases concern external steady or unsteady flows around a circular cylinder, heated or not, and the channel flow behind a backward‐facing step. The numerical results are shown in good agreement with other published numerical or experimental data. Copyright © 2000 John Wiley & Sons, Ltd.     

Three‐dimensional incompressible flow calculations using the characteristic based split (CBS) scheme

In this paper, the characteristic based split scheme is employed for the solution of three‐dimensional incompressible viscous flow problems on unstructured meshes. Many algorithm related issues are discussed. Fully explicit and semiimplicit forms of the scheme are explained and employed in the calculation of both isothermal and nonisothermal incompressible flows simulation. The extension of the scheme to porous medium flows is also demonstrated with relevant examples. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid pressure‐based solver for nonideal single‐phase fluid flows at all speeds

This paper describes the implementation of a numerical solver that is capable of simulating compressible flows of nonideal single‐phase fluids. The proposed method can be applied to arbitrary equations of state and is suitable for all Mach numbers. The pressure‐based solver uses the operator‐splitting technique and is based on the PISO/SIMPLE algorithm: the density, velocity, and temperature fields are predicted by solving the linearized versions of the balance equations using the convective fluxes from the previous iteration or time step. The overall mass continuity is ensured by solving the pressure equation derived from the continuity equation, the momentum equation, and the equation of state. Nonphysical oscillations of the numerical solution near discontinuities are damped using the Kurganov‐Tadmor/Kurganov‐Noelle‐Petrova (KT/KNP) scheme for convective fluxes. The solver was validated using different test cases, where analytical and/or numerical solutions are present or can be derived: (1) A convergent‐divergent nozzle with three different operating conditions; (2) the Riemann problem for the Peng‐Robinson equation of state; (3) the Riemann problem for the covolume equation of state; (4) the development of a laminar velocity profile in a circular pipe (also known as Poiseuille flow); (5) a laminar flow over a circular cylinder; (6) a subsonic flow over a backward‐facing step at low Reynolds numbers; (7) a transonic flow over the RAE 2822 airfoil; and (8) a supersonic flow around a blunt cylinder‐flare model. The spatial approximation order of the scheme is second order. The mesh convergence of the numerical solution was achieved for all cases. The accuracy order for highly compressible flows with discontinuities is close to first order and, for incompressible viscous flows, it is close to second order. The proposed solver is named rhoPimpleCentralFoam and is implemented in the open‐source CFD library OpenFOAM^®. For high speed flows, it shows a similar behavior as the KT/KNP schemes (implemented as rhoCentralFoam‐solver, Int. J. Numer. Meth. Fluids 2010), and for flows with small Mach numbers, it behaves like solvers that are based on the PISO/SIMPLE algorithm.     

Numerical implementation of Aristov–Pukhnachev's formulation for axisymmetric viscous incompressible flows

In the present work a finite‐difference technique is developed for the implementation of a new method proposed by Aristov and Pukhnachev (Doklady Phys. 2004; 49 (2):112–115) for modeling of the axisymmetric viscous incompressible fluid flows. A new function is introduced that is related to the pressure and a system similar to the vorticity/stream function formulation is derived for the cross‐flow. This system is coupled to an equation for the azimuthal velocity component. The scheme and the algorithm treat the equations for the cross‐flow as an inextricably coupled system, which allows one to satisfy two conditions for the stream function with no condition on the auxiliary function. The issue of singularity of the matrix is tackled by adding a small parameter in the boundary conditions. The scheme is thoroughly validated on grids with different resolutions. The new numerical tool is applied to the Taylor flow between concentric rotating cylinders when the upper and lower lids are allowed to rotate independently from the inner cylinder, while the outer cylinder is held at rest. The phenomenology of this flow is adequately represented by the numerical model, including the hysteresis that takes place near certain specific values of the Reynolds number. Thus, the present results can be construed to demonstrate the viability of the new model. The success can be attributed to the adequate physical nature of the auxiliary function. The proposed technique can be used in the future for in‐depth investigations of the bifurcation phenomena in rotating flows. Copyright © 2009 John Wiley & Sons, Ltd.     

The development of a Cartesian cut cell method for incompressible viscous flows

This paper describes the extension of the Cartesian cut cell method to applications involving unsteady incompressible viscous fluid flow. The underlying scheme is based on the solution of the full Navier–Stokes equations for a variable density fluid system using the artificial compressibility technique together with a Jameson‐type dual time iteration. The computational domain encompasses two fluid regions and the interface between them is treated as a contact discontinuity in the density field, thereby eliminating the need for special free surface tracking procedures. The Cartesian cut cell technique is used for fitting the complex geometry of solid boundaries across a stationary background Cartesian grid which is located inside the computational domain. A time accurate solution is achieved by using an implicit dual‐time iteration technique based on a slope‐limited, high‐order, Godunov‐type scheme for the inviscid fluxes, while the viscous fluxes are estimated using central differencing. Validation of the new technique is by modelling the unsteady Couette flow and the Rayleigh–Taylor instability problems. Finally, a test case for wave run‐up and overtopping over an impermeable sea dike is performed. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite‐volume‐type VOF method on dynamically adaptive quadtree grids

This paper describes a finite‐volume volume‐of‐fluid (VOF) method for simulating viscous free surface flows on dynamically adaptive quadtree grids. The scheme is computationally efficient in that it provides relatively fine grid resolution at the gas–liquid interface and coarse grid density in regions where flow variable gradients are small. Special interpolations are used to ensure volume flux conservation where differently sized neighbour cells occur. The numerical model is validated for advection of dyed fluid in unidirectional and rotating flows, and for two‐dimensional viscous sloshing in a rectangular tank. Copyright © 2004 John Wiley & Sons, Ltd.